Why does render to texture camera make armature animation freeze in the Game engine?
There is a "animation culling" build in. This means anything that is not seen by the active camera will not be animated. This is to skip processing time.

This is not always a correct assumption, e.g. when the animation influences physics or as in your case you see it by a different camera. 

I know this issue was raised as a bug to allow disabling this sort of culling for certain objects. Just now I do not know the state of that feature.